Chapter 3 The K_0 Group of a Unital C* -algebra.
An abelian group
The Grothendieck construction 3.1.1
One can associate an abelian group to every abelian semigroup in a way analogous to how one obtains the integers from the naturals, and in much the same way as one obtains the rationals from the integers. We describe here how this works; the proofs of various statements are deferred to the next paragraph.
Let
That
Write
The operation
is well-defined and turns
The group
Let
Properties of the Grothendieck group
The Grothendieck construction has the following properties.
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Let
So take any projection p, and send it to its K_0 class via the Grothendieck map on its equivalence class inside of the semigroup of projections .
The group K_00 3.1.5
The definition of
For unital
Definition 3.1.6 (Stable equivalence)
Define a relation
Characterization for stable equivalence
If
Proof:
If
The standard picture of
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Now we proof 1:
We know that
Proof of 2:
We use the fact that
Proof of 3:
Follows from the strength of homotopy over Murray von Neumann equivalence.
Proof of 4:
We have that
Proof of 5:
If
Hence
Conversely, if
Proposition 3.1.8 (Universal property of K0)
Let
The functor K0 for unital C-star Algebras 3.2.2The functor K00 3.2.3
Let
By Section 1.3,
Define
If
This raises an important question, is this new functor K00, still not half- exact? we had noted that it was a brutal flaw of defining K0 for non-unital algebras. later in example 3.3.9 it will be proven that K00 is not half-exact…. but then in chapter 4 we get a new definition of K0 and K00 using the unitization, so should this paragraph above be ignored altogether?
Proposition 3.2.6 (Homotopy invariance of K0)
Let
Proof:
Two
#mutually_orthogonal_star_hom
Lemma 3.2.7
If
Lemma 3.2.8
For every unital
Paragraph 3.3.1 Traces and K0
Let
A trace is positive if it maps positive elements to positive elements.
If a trace is positive and
Any trace
this trace on the matrix algebra then gives rise to a function
Example 3.3.2
The group
This fact implies that the 0th k-theory of
Example 3.3.3
Let
Example 3.3.5
Let